Vol. 11, No. 7, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Quantitative stability of the free boundary in the obstacle problem

Sylvia Serfaty and Joaquim Serra

Vol. 11 (2018), No. 7, 1803–1839
Abstract

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in n (n 2) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field.

To do so, working in the setting of the whole space, we examine the evolution of the free boundary Γt corresponding to the boundary of the contact set for a family of obstacle functions ht . Assuming that h = ht(x) = h(t,x) is Ck+1,α in [1,1] × n and that the initial free boundary Γ0 is regular, we prove that Γt is twice differentiable in t in a small neighborhood of t = 0. Moreover, we show that the “normal velocity” and the “normal acceleration” of Γt are respectively Ck1,α and Ck2,α scalar fields on Γt . This is accomplished by deriving equations for this velocity and acceleration and studying the regularity of their solutions via single- and double-layer estimates from potential theory.

Keywords
obstacle problem, contact set, coincidence set, stability, equilibrium measure, potential theory
Mathematical Subject Classification 2010
Primary: 35R35, 31B35, 49K99
Milestones
Received: 10 August 2017
Revised: 13 February 2018
Accepted: 9 April 2018
Published: 20 May 2018
Authors
Sylvia Serfaty
Courant Institute of Mathematical Sciences, New York University
New York, NY
United States
Joaquim Serra
Department of Mathematics
ETH Zürich
Zürich
Switzerland